Physics for Science & Engineering II
Physics for Science & Engineering II
By Yildirim Aktas, Department of Physics & Optical Science
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  • Introduction
  • Syllabus
  • Online Lectures
    • Chapter 01: Electric Charge
      • 1.1 Fundamental Interactions
      • 1.2 Electrical Interactions
      • 1.3 Electrical Interactions 2
      • 1.4 Properties of Charge
      • 1.5 Conductors and Insulators
      • 1.6 Charging by Induction
      • 1.7 Coulomb Law
        • Example 1: Equilibrium Charge
        • Example 2: Three Point Charges
        • Example 3: Charge Pendulums
    • Chapter 02: Electric Field
      • 2.1 Electric Field
      • 2.2 Electric Field of a Point Charge
      • 2.3 Electric Field of an Electric Dipole
      • 2.4 Electric Field of Charge Distributions
        • Example 1: Electric field of a charged rod along its Axis
        • Example 2: Electric field of a charged ring along its axis
        • Example 3: Electric field of a charged disc along its axis
        • Example 4: Electric field of a charged infinitely long rod.
        • Example 5: Electric field of a finite length rod along its bisector.
      • 2.5 Dipole in an External Electric Field
    • Chapter 03: Gauss’ s Law
      • 3.1 Gauss’s Law
        • Example 1: Electric field of a point charge
        • Example 2: Electric field of a uniformly charged spherical shell
        • Example 3: Electric field of a uniformly charged soild sphere
        • Example 4: Electric field of an infinite, uniformly charged straight rod
        • Example 5: Electric Field of an infinite sheet of charge
        • Example 6: Electric field of a non-uniform charge distribution
      • 3.2 Conducting Charge Distributions
        • Example 1: Electric field of a concentric solid spherical and conducting spherical shell charge distribution
        • Example 2: Electric field of an infinite conducting sheet charge
      • 3.3 Superposition of Electric Fields
        • Example: Infinite sheet charge with a small circular hole.
    • Chapter 04: Electric Potential
      • 4.1 Potential
      • 4.2 Equipotential Surfaces
        • Example 1: Potential of a point charge
        • Example 2: Potential of an electric dipole
        • Example 3: Potential of a ring charge distribution
        • Example 4: Potential of a disc charge distribution
      • 4.3 Calculating potential from electric field
      • 4.4 Calculating electric field from potential
        • Example 1: Calculating electric field of a disc charge from its potential
        • Example 2: Calculating electric field of a ring charge from its potential
      • 4.5 Potential Energy of System of Point Charges
      • 4.6 Insulated Conductor
    • Chapter 05: Capacitance
      • 5.01 Introduction
      • 5.02 Capacitance
      • 5.03 Procedure for calculating capacitance
      • 5.04 Parallel Plate Capacitor
      • 5.05 Cylindrical Capacitor
      • 5.06 Spherical Capacitor
      • 5.07-08 Connections of Capacitors
        • 5.07 Parallel Connection of Capacitors
        • 5.08 Series Connection of Capacitors
          • Demonstration: Energy Stored in a Capacitor
          • Example: Connections of Capacitors
      • 5.09 Energy Stored in Capacitors
      • 5.10 Energy Density
      • 5.11 Example
    • Chapter 06: Electric Current and Resistance
      • 6.01 Current
      • 6.02 Current Density
        • Example: Current Density
      • 6.03 Drift Speed
        • Example: Drift Speed
      • 6.04 Resistance and Resistivity
      • 6.05 Ohm’s Law
      • 6.06 Calculating Resistance from Resistivity
      • 6.07 Example
      • 6.08 Temperature Dependence of Resistivity
      • 6.09 Electromotive Force, emf
      • 6.10 Power Supplied, Power Dissipated
      • 6.11 Connection of Resistances: Series and Parallel
        • Example: Connection of Resistances: Series and Parallel
      • 6.12 Kirchoff’s Rules
        • Example: Kirchoff’s Rules
      • 6.13 Potential difference between two points in a circuit
      • 6.14 RC-Circuits
        • Example: 6.14 RC-Circuits
    • Chapter 07: Magnetism
      • 7.1 Magnetism
      • 7.2 Magnetic Field: Biot-Savart Law
        • Example: Magnetic field of a current loop
        • Example: Magnetic field of an infinitine, straight current carrying wire
        • Example: Semicircular wires
      • 7.3 Ampere’s Law
        • Example: Infinite, straight current carrying wire
        • Example: Magnetic field of a coaxial cable
        • Example: Magnetic field of a perfect solenoid
        • Example: Magnetic field of a toroid
        • Example: Magnetic field profile of a cylindrical wire
        • Example: Variable current density
    • Chapter 08: Magnetic Force
      • 8.1 Magnetic Force
      • 8.2 Motion of a charged particle in an external magnetic field
      • 8.3 Current carrying wire in an external magnetic field
      • 8.4 Torque on a current loop
      • 8.5 Magnetic Domain and Electromagnet
      • 8.6 Magnetic Dipole Energy
      • 8.7 Current Carrying Parallel Wires
        • Example 1: Parallel Wires
        • Example 2: Parallel Wires
    • Chapter 09: Induction
      • 9.1 Magnetic Flux, Fraday’s Law and Lenz Law
        • Example: Changing Magnetic Flux
        • Example: Generator
        • Example: Motional emf
        • Example: Terminal Velocity
        • Simulation: Faraday’s Law
      • 9.2 Induced Electric Fields
      • Inductance
        • 9.3 Inductance
        • 9.4 Procedure to Calculate Inductance
        • 9.5 Inductance of a Solenoid
        • 9.6 Inductance of a Toroid
        • 9.7 Self Induction
        • 9.8 RL-Circuits
        • 9.9 Energy Stored in Magnetic Field and Energy Density
      • Maxwell’s Equations
        • 9.10 Maxwell’s Equations, Integral Form
        • 9.11 Displacement Current
        • 9.12 Maxwell’s Equations, Differential Form
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Online Lectures » Chapter 09: Induction » Inductance » 9.5 Inductance of a Solenoid

9.5 Inductance of a Solenoid

from Office of Academic Technologies on Vimeo.

9.5 Inductance of a Solenoid

Let’s assume that we have a solenoid, something like this, with N number of turns, and we let current i flow through this solenoid, and as a result of that we generate a magnetic field B, filling the region surrounded by the solenoid. We’d like to calculate the inductance of the solenoid, and therefore we will follow the procedure that we have introduced earlier.

First, we assume that the certain amount of current is flowing through the solenoid. As the second step, we will calculate the magnetic field of this solenoid. We will assume that the solenoid that we’re dealing with is a perfect solenoid, and let’s assume that its total length is given as l, length of solenoid, and let’s assume that little n, which is equal to number of turns per unit length, gives us the number density of the solenoid, number density of turns.

All right. In order to calculate the magnetic field, we’re going to apply Ampere’s Law. As a matter of fact, we did this earlier step by step, but let’s do it one more time as a review. If we just cut the solenoid vertically down, we end up with an upper branch and a lower branch, and if the upper branch is carrying the current out of plane direction, then the lower branch will carry the current into the plane direction. In other words, if the current is coming out of here, then it is going to go into the plane here and then it will come out here and go into the plane there, and so on and so forth.

In such a current configuration as we have seen earlier, it’s going to generate a magnetic field along the axis of the solenoid, and fill the interior region of the solenoid. Now, for a perfect solenoid, as you recall, this is the case that l is much, much greater than the, let’s say, diameter of the solenoid. Then one can assume that Bout, the magnetic field outside of the solenoid, is 0. Of course, this is not the actual case, but comparing the strength of the solenoid inside relative to the one outside, one can actually make this approximation.

Let’s assume that we’re interested with the magnetic field at this point P, an arbitrary point inside of the solenoid, and to be able to calculate the magnetic field, first we look at the field line passing through that point. So it’s going to be something like this. And we’re going to choose a closed loop such that one side of this loop will coincide with the magnetic field line passing through the point of interest.

So, we’ll choose a rectangular loop in this case, of this shape, and if we just say this side is the segment one and this one is segment two, segment three, and segment four, the Ampere’s Law, which says integral of B dot dl over a closed contour C is equal to μ0 times i-enclosed. This closed loop integral can be expressed as the open path integrals of the four sides of this rectangular loop, and then they are added together to be able to get the whole closed loop integral.

In other words, we can express this as integral over the first segment of B dot dl plus integral over the second segment of B dot dl plus integral over the third segment of B dot dl and plus  integral over the fourth segment of B dot dl. They all added to closed path integral or closed loop integral and will be equal to μ0 times i-enclosed.

Now, if we look at each one of these segments separately, for the first segment, dl is pointing down and magnetic field is pointing to the right. Therefore the angle between them will be 90 degrees inside of the solenoid. If we look at the outside of the solenoid, here Bout is 0, therefore from the outside part of the first segment, since magnetic field is 0, there will not be any contribution, and for the inside region of the first segment, the angle between B and dl is 90 degrees, so this is going to give us 0. And reason, that the

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